Propagating waves and edge vibrations in anisotropic composite cylinders

Abstract

The entire dispersive spectra of a cylinder with cylindrical anisotropy are determined from three different algebraic eigenvalue problems deducible from the same finite element formulation. The displacement vector v in this version of the finite element method has the form f( r) exp i( εz + nθ + ωt) with the radial dependence f( r) taken as quadratic interpolation polynomials. Therefore, this discretization procedure allows a cylinder with radially inhomogeneous material properties to be modeled. The three different algebraic eigenvalue problems that emerge depend on whether the axial wave number ε or the natural frequency ω is regarded as the eigenvalue parameter and on the real, purely imaginary or complex nature of ε. For ε specified as real, an eigenvalue problem results for the natural frequencies ω i for waves propagating along the z-direction of a cylinder of infinite extent. When ε is specified to be purely imaginary, then an algebraic eigenvalue problem governing the edge vibrations (end modes) of a semi-infinite cylinder is obtained. The third eigenvalue problem can be obtained by considering ω to be prescribed and regarding ε as the eigenvalue parameter. The algebraic eigenvalue problem that results is quadratic in the eigenvalue parameter and admits solutions for ε which may be real, purely imaginary or complex. Complex ε's correspond to edge vibrations in a cylinder which are exponentially damped trigonometric wave forms. Moreover, for the case ω = 0, the eigenvalue analysis yields ε as the characteristic inverse decay lengths for systems of elastostatic self-equilibrated edge effects in the context of St. Venant's principle. All the eigenvalue problems are solved by efficient techniques based on subspace iteration. Examples of two four-layer angle-ply cylinders are presented to illustrate this comprehensive finite element analysis.